An observer finds that the angular elevation of a tower is `theta`. On advancing 3m towards the tower, the elevation is `45^(@)` and on advancing 2m further more towards the tower, the elevation is `90^(@)-theta`. The height of the tower is (assume the height of observer is negligible and observer lies on the same level as the foot of the tower)

Options

A. 2m
B. 4m
C. 6m
D. 8m

Answer

To solve the problem step by step, we will use trigonometric relationships based on the angles of elevation given.

### Step 1: Set Up the Problem
Let the height of the tower be \( H \) meters. The observer starts at a point \( C \), moves 3 meters towards the tower to point \( D \), and then moves 2 meters further to point \( E \).

### Step 2: Define Distances
Let the distance from the base of the tower to point \( C \) be \( x + 5 \) meters (where \( x \) is the distance from the base of the tower to point \( E \)). After moving 3 meters to point \( D \), the distance to the tower is \( x + 2 \) meters. After moving another 2 meters to point \( E \), the distance to the tower is \( x \) meters.

### Step 3: Use the Angle of Elevation at Point D
At point \( D \), the angle of elevation is \( 45^\circ \). Using the tangent function:
\[ \tan(45^\circ) = \frac{H}{x + 2} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{H}{x + 2} \implies H = x + 2 \quad \text{(1)} \]

### Step 4: Use the Angle of Elevation at Point C
At point \( C \), the angle of elevation is \( \theta \). Using the tangent function:
\[ \tan(\theta) = \frac{H}{x + 5} \]
Thus:
\[ H = (x + 5) \tan(\theta) \quad \text{(2)} \]

### Step 5: Use the Angle of Elevation at Point E
At point \( E \), the angle of elevation is \( 90^\circ - \theta \). Using the cotangent function:
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{H}{x} \]
Thus:
\[ H = x \cot(\theta) \quad \text{(3)} \]

### Step 6: Equate the Expressions for H
From equations (1), (2), and (3):
1. From (1): \( H = x + 2 \)
2. From (2): \( H = (x + 5) \tan(\theta) \)
3. From (3): \( H = x \cot(\theta) \)

### Step 7: Substitute and Solve
Set equations (1) and (2) equal:
\[ x + 2 = (x + 5) \tan(\theta) \]
Expanding gives:
\[ x + 2 = x \tan(\theta) + 5 \tan(\theta) \]
Rearranging:
\[ x - x \tan(\theta) = 5 \tan(\theta) - 2 \]
Factoring out \( x \):
\[ x(1 - \tan(\theta)) = 5 \tan(\theta) - 2 \]
Thus:
\[ x = \frac{5 \tan(\theta) - 2}{1 - \tan(\theta)} \quad \text{(4)} \]

Now set equations (1) and (3) equal:
\[ x + 2 = x \cot(\theta) \]
Rearranging gives:
\[ x \cot(\theta) - x = 2 \implies x(\cot(\theta) - 1) = 2 \]
Thus:
\[ x = \frac{2}{\cot(\theta) - 1} \quad \text{(5)} \]

### Step 8: Equate (4) and (5)
Set the two expressions for \( x \) equal:
\[ \frac{5 \tan(\theta) - 2}{1 - \tan(\theta)} = \frac{2}{\cot(\theta) - 1} \]
Cross-multiplying and simplifying will yield a relationship involving \( H \) and \( \theta \).

### Step 9: Solve for H
After simplifying the above equation, you will find that:
\[ H = 6 \]

### Final Answer
The height of the tower is \( H = 6 \) meters.

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